Abstract

The polarization mode dispersion (PMD) of a nondepolarizing optical system is analyzed. A new method is proposed to calculate the complex PMD vector by measuring two Jones matrices at two different wavelengths.

© 2007 Optical Society of America

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References

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  1. R. Simon, "Nondepolarizing systems and degree of polarization," Opt. Commun. 77, 349-354 (1990).
    [CrossRef]
  2. N. Gisin and B. Hutter, "Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers," Opt. Commun. 142, 119-125 (1997).
    [CrossRef]
  3. Liang Chen, Ou Chen, S. Hadjifaradji, and Xiaoyi Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16, 206-208 (2004).
    [CrossRef]
  4. R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999).
    [CrossRef]
  5. A. Eyal and M. Tur, "Measurement of polarization mode dispersion in systems having polarization dependent loss or gain," IEEE Photon. Technol. Lett. 9, 1256-1258 (1997).
    [CrossRef]
  6. R. C. Jones, "A new calculus for the treatment of optical systems. VI. Experimental determination of the matrix," J. Opt. Soc. Am. 37, 110-112 (1947).
    [CrossRef]
  7. P. Conrad, Differential Equations: a Systems Approach, 1st ed. (Prentice-Hall, 2003), p. 176.
  8. B. L. Heffner, "Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis," IEEE Photon. Technol. Lett. 4, 1066-1069 (1992).
    [CrossRef]
  9. R. E. Larson and B. H. Edwards, Elementary Linear Algebra, 2nd ed. (Heath, 1991), pp. 385-389.
  10. M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, 1974).
  11. C. S. Kenney and A. J. Laub, "A Schur-Frechet algorithm for computing the logarithm and exponential of a matrix," Soc. Indust. Appl. Math. 19, 640-663 (1998).
  12. A. Bessa dos Santo and J. P. von der weid, "PDL effects in PMD emulators made out with HiBi fibers: building PMD/PDL emulators," IEEE Photon. Technol. Lett. 16, 452-454 (2004).
    [CrossRef]
  13. B. L. Heffner, "Deterministic, analytically complete measurement of polarization-dependent transmission through optical devices," IEEE Photon. Technol. Lett. 4, 451-454 (1992).
    [CrossRef]

2004 (2)

Liang Chen, Ou Chen, S. Hadjifaradji, and Xiaoyi Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16, 206-208 (2004).
[CrossRef]

A. Bessa dos Santo and J. P. von der weid, "PDL effects in PMD emulators made out with HiBi fibers: building PMD/PDL emulators," IEEE Photon. Technol. Lett. 16, 452-454 (2004).
[CrossRef]

1999 (1)

R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999).
[CrossRef]

1998 (1)

C. S. Kenney and A. J. Laub, "A Schur-Frechet algorithm for computing the logarithm and exponential of a matrix," Soc. Indust. Appl. Math. 19, 640-663 (1998).

1997 (2)

N. Gisin and B. Hutter, "Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers," Opt. Commun. 142, 119-125 (1997).
[CrossRef]

A. Eyal and M. Tur, "Measurement of polarization mode dispersion in systems having polarization dependent loss or gain," IEEE Photon. Technol. Lett. 9, 1256-1258 (1997).
[CrossRef]

1992 (2)

B. L. Heffner, "Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis," IEEE Photon. Technol. Lett. 4, 1066-1069 (1992).
[CrossRef]

B. L. Heffner, "Deterministic, analytically complete measurement of polarization-dependent transmission through optical devices," IEEE Photon. Technol. Lett. 4, 451-454 (1992).
[CrossRef]

1990 (1)

R. Simon, "Nondepolarizing systems and degree of polarization," Opt. Commun. 77, 349-354 (1990).
[CrossRef]

1947 (1)

Bao, Xiaoyi

Liang Chen, Ou Chen, S. Hadjifaradji, and Xiaoyi Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16, 206-208 (2004).
[CrossRef]

Bessa dos Santo, A.

A. Bessa dos Santo and J. P. von der weid, "PDL effects in PMD emulators made out with HiBi fibers: building PMD/PDL emulators," IEEE Photon. Technol. Lett. 16, 452-454 (2004).
[CrossRef]

Chen, Liang

Liang Chen, Ou Chen, S. Hadjifaradji, and Xiaoyi Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16, 206-208 (2004).
[CrossRef]

Chen, Ou

Liang Chen, Ou Chen, S. Hadjifaradji, and Xiaoyi Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16, 206-208 (2004).
[CrossRef]

Conrad, P.

P. Conrad, Differential Equations: a Systems Approach, 1st ed. (Prentice-Hall, 2003), p. 176.

Edwards, B. H.

R. E. Larson and B. H. Edwards, Elementary Linear Algebra, 2nd ed. (Heath, 1991), pp. 385-389.

Eyal, A.

A. Eyal and M. Tur, "Measurement of polarization mode dispersion in systems having polarization dependent loss or gain," IEEE Photon. Technol. Lett. 9, 1256-1258 (1997).
[CrossRef]

Gisin, N.

N. Gisin and B. Hutter, "Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers," Opt. Commun. 142, 119-125 (1997).
[CrossRef]

Hadjifaradji, S.

Liang Chen, Ou Chen, S. Hadjifaradji, and Xiaoyi Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16, 206-208 (2004).
[CrossRef]

Heffner, B. L.

B. L. Heffner, "Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis," IEEE Photon. Technol. Lett. 4, 1066-1069 (1992).
[CrossRef]

B. L. Heffner, "Deterministic, analytically complete measurement of polarization-dependent transmission through optical devices," IEEE Photon. Technol. Lett. 4, 451-454 (1992).
[CrossRef]

Hirsch, M. W.

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, 1974).

Hutter, B.

N. Gisin and B. Hutter, "Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers," Opt. Commun. 142, 119-125 (1997).
[CrossRef]

Jones, R. C.

Jopson, R. M.

R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999).
[CrossRef]

Kenney, C. S.

C. S. Kenney and A. J. Laub, "A Schur-Frechet algorithm for computing the logarithm and exponential of a matrix," Soc. Indust. Appl. Math. 19, 640-663 (1998).

Kogelnik, H.

R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999).
[CrossRef]

Larson, R. E.

R. E. Larson and B. H. Edwards, Elementary Linear Algebra, 2nd ed. (Heath, 1991), pp. 385-389.

Laub, A. J.

C. S. Kenney and A. J. Laub, "A Schur-Frechet algorithm for computing the logarithm and exponential of a matrix," Soc. Indust. Appl. Math. 19, 640-663 (1998).

Nelson, L. E.

R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999).
[CrossRef]

Simon, R.

R. Simon, "Nondepolarizing systems and degree of polarization," Opt. Commun. 77, 349-354 (1990).
[CrossRef]

Smale, S.

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, 1974).

Tur, M.

A. Eyal and M. Tur, "Measurement of polarization mode dispersion in systems having polarization dependent loss or gain," IEEE Photon. Technol. Lett. 9, 1256-1258 (1997).
[CrossRef]

von der weid, J. P.

A. Bessa dos Santo and J. P. von der weid, "PDL effects in PMD emulators made out with HiBi fibers: building PMD/PDL emulators," IEEE Photon. Technol. Lett. 16, 452-454 (2004).
[CrossRef]

IEEE Photon. Technol. Lett. (6)

Liang Chen, Ou Chen, S. Hadjifaradji, and Xiaoyi Bao, "Polarization-mode dispersion measurement in a system with polarization-dependent loss or gain," IEEE Photon. Technol. Lett. 16, 206-208 (2004).
[CrossRef]

R. M. Jopson, L. E. Nelson, and H. Kogelnik, "Measurement of second-order polarization-mode dispersion vectors in optical fibers," IEEE Photon. Technol. Lett. 11, 1153-1155 (1999).
[CrossRef]

A. Eyal and M. Tur, "Measurement of polarization mode dispersion in systems having polarization dependent loss or gain," IEEE Photon. Technol. Lett. 9, 1256-1258 (1997).
[CrossRef]

B. L. Heffner, "Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis," IEEE Photon. Technol. Lett. 4, 1066-1069 (1992).
[CrossRef]

A. Bessa dos Santo and J. P. von der weid, "PDL effects in PMD emulators made out with HiBi fibers: building PMD/PDL emulators," IEEE Photon. Technol. Lett. 16, 452-454 (2004).
[CrossRef]

B. L. Heffner, "Deterministic, analytically complete measurement of polarization-dependent transmission through optical devices," IEEE Photon. Technol. Lett. 4, 451-454 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

R. Simon, "Nondepolarizing systems and degree of polarization," Opt. Commun. 77, 349-354 (1990).
[CrossRef]

N. Gisin and B. Hutter, "Combined effects of polarization mode dispersion and polarization dependent losses in optical fibers," Opt. Commun. 142, 119-125 (1997).
[CrossRef]

Soc. Indust. Appl. Math. (1)

C. S. Kenney and A. J. Laub, "A Schur-Frechet algorithm for computing the logarithm and exponential of a matrix," Soc. Indust. Appl. Math. 19, 640-663 (1998).

Other (3)

R. E. Larson and B. H. Edwards, Elementary Linear Algebra, 2nd ed. (Heath, 1991), pp. 385-389.

M. W. Hirsch and S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, 1974).

P. Conrad, Differential Equations: a Systems Approach, 1st ed. (Prentice-Hall, 2003), p. 176.

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Figures (5)

Fig. 1
Fig. 1

Experimental setup to measure PMD.

Fig. 2
Fig. 2

(a) Real part Ω = Ω 1 a x + Ω 2 a y + Ω 3 a z and (b) the imaginary part Λ = Λ 1 a x + Λ 2 a y + Λ 3 a z of the complex PMD vector W = Ω + i Λ versus wavelength.

Fig. 3
Fig. 3

DGD measurement using three methods: Jones matrix eigenanalysis method (JME), complex plane method (CP), and generalized Jones matrix analysis method (GJMA).

Fig. 4
Fig. 4

DAS of the optical system versus wavelength.

Fig. 5
Fig. 5

PDL of the optical system versus wavelength.

Equations (30)

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T T 1 = j 2 W σ A ,
σ x = [ 1 0 0 1 ] , σ y = [ 0 1 1 0 ] , σ z = [ 0 j j 0 ] .
τ D G D = Re ( W W ) .
η D A S = Im ( W W ) .
D G D = Im { ( A 22 A 11 ) 2 + 4 A 12 A 21 } .
W σ φ ± = τ ± φ ± ,
DGD = Re { τ } ,
χ ± = 1 2 A 12 [ ( A 22 A 11 ) ± ( A 22 A 11 ) 2 + 4 A 12 A 21 ] .
k 1 = h x / h y , k 2 = v x / v y , k 3 = q x / q y ,
k 4 = ( k 3 k 2 ) / ( k 1 k 3 ) ,
T = β [ k 1 k 4 k 2 k 4 1 ] β N ,
T = A T .
[ T 1 i T 2 i ] = exp ( A ω ) [ C 1 i C 2 i ] , i = 1 , 2   or
T = exp ( A ω ) C ,
T 2 T 1 1 = exp ( A ω 2 ) exp ( A ω 1 ) = exp [ A ( ω 2 ω 1 ) ]
A = ln T 2 T 1 1 ω 2 ω 1 .
A = ln β 2 β 1 1 N 2 N 1 1 ω 2 ω 1 = I ln β 2 β 1 1 + ln N 2 N 1 1 ω 2 ω 1 a I + V ,
j 2 W σ = a I + ln N 2 N 1 1 ω 2 ω 1 .
A φ ± = j τ ± 2 φ ± .
A = P [ j τ 2 0 0 j τ 2 ] P 1 .
exp ( A ) = k = 0 A k k ! ,
T = P [ exp ( j τ ω / 2 ) 0 0 exp ( j τ ω / 2 ) ] P 1 C .
T 2 T 1 1 = exp ( A Δ ω ) = P [ exp ( j τ Δ ω / 2 ) 0 0 exp ( j τ Δ ω / 2 ) ] P 1 .
N 2 N 1 1 = β 1 β 2 1 T 2 T 1 1 P [ exp ( j τ Δ ω / 2 + φ ) 0 0 exp ( j τ Δ ω / 2 + φ ) ] P 1 ,
ln { N 2 N 1 1 } = P [ ln exp ( j τ Δ ω / 2 + φ ) 0 0 ln exp ( j τ Δ ω / 2 + φ ) ] P 1 .
Im [ g [ Ln { N 2 N 1 1 } ] ] = | Im { ± j τ Δ ω / 2 + φ } | π .
| | a | π ± | b | π | π = | a ± b | π .
| Im { g + g } | π = | | Im { j τ Δ ω / 2 + φ } | π | Im { j τ Δ ω / 2 + φ } | π | π = | Im { j τ Δ ω } | π = | DGD Δ ω | π ,
DGD = | Im { g + g } | π Δ ω .
Δ ω D G D corrected = | Δ ω D G D uncorrected | π ,

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